Mediation and Moderation:

Mediation and Moderation:
What are they?
Are they for me?
0011 0010 1010 1101 0001 0100 1011
1
2
4
Overview
• Definitions of M&M and Differences
between M&M
• Methods for testing for mediation
• Advanced topics in mediation
• Methods for testing for moderation
• Advanced topics in moderation
• Advanced topics in mediation and
moderation
Simple Mediation and Moderation:
Similarities
• Both involve 3 variables
• The goal is to determine how a third
variable affects the simple relationship
between two variables
• Both can be analyzed with regression or
structural equation modeling (SEM)
software
Simple Mediation and Moderation:
Differences
• Moderation involves product terms, not
mediation
• Mediation is best used with longitudinal
data, but this is not required for moderation
• Centering variables is usually required in
moderation, but not in mediation
• Graphing is essential in moderation, but
only highly recommended in mediation
Definitions and
Important Questions
• Mediation
– A mediating variable is the mechanism by
which an effect occurs between a predictor and
an outcome
– The important question is whether a third
variable (Med) partially or completely controls
the relationship between a predictor (X) and an
outcome (Y)
Definitions and
Important Questions
• Moderation
– A moderating variable is one that affects the
direction and/or strength of the relationship
between a predictor variable and a criterion
variable
– The important question is whether the
relationship between the predictor (X) and the
outcome (Y) differs across the levels of the
moderator (mod)
Frequency of Mediation Analyses
Frequency
• Number of mentions of “mediation”
“mediator” or “mediates” in the title of
psychology journal articles, 1970 - 2010
3000
2500
2000
1500
1000
500
0
2893
1182
267
1970s
639
1980s
1990s
Years
2000s
Introduction to Mediation
• Simple Model
Introduction to Mediation
• Mediation Model
Introduction to Mediation
• Rephrasing the important question:
– Does the simple relationship between X and Y
[c] shrink or disappear when the mediator is
present in the model (i.e., is c΄<c, or is c΄=0)?
– A relationship that shrinks indicates partial
mediation, whereas a relationship that
disappears indicates full mediation
• More on full and partial mediation to come
Decomposition of Effects
• Total Effect = Indirect Effect + Direct Effect
• c = c΄ + ab
– Thus, ab = c - c΄
• This holds exactly for least squares estimates
(e.g., standard multiple regression), but is
only an approximation with other estimation
methods
Baron & Kenny …
I wish I was cited like that
• The most famous reference to mediation is
Baron & Kenny (1986)
• Baron, R. M., & Kenny, D. A. (1986). The moderatormediator variable distinction in social psychological
research: Conceptual, strategic, and statistical
considerations. Journal of Personality and Social
Psychology, 51, 1173-1182
• This paper has been referenced more than
25000 times
Why was the Baron & Kenny
article so popular?
• It laid out mediation in a set of simple to
implement steps:
1)
2)
3)
4)
Test X -> Y (path c)
Test X -> M (path a)
Test M -> Y, controlling for X (path b)
Test X -> Y, controlling for M (path c΄)
– If steps 1 to 3 are significant, and c΄ is reduced
then we have partial mediation (or if c΄ = 0 we
have full mediation)
Criticisms of the Baron & Kenny
Steps
• The main criticisms of the Baron & Kenny
approach relate to the necessity of the steps
• A significant X to Y path is not required (e.g., the
direct and indirect effect may be of different signs)
• Other researchers have argued that the M to Y path
may not be significant when X and M are highly
correlated (multicollinearity)
• Further, a statistical test of X -> Y in the last step is
meaningless since it cannot prove partial or full
mediation
How do we test for partial
mediation?
• There are two popular approaches for
testing whether c΄ < c (i.e., whether partial
mediation has occurred)
– Sobel Test
• OK test when sample sizes are large (>100)
– Bootstrapping
• Good test for small or large sample sizes
Sobel Test
• The Sobel test assesses the statistical
significance of the indirect effect (ab)
• In other words, it assesses whether a significant part
of the relationship between X and Y is controlled by
the mediator
– Let sa and sb represent the standard errors of a
and b, and reject Ho: ab = 0 if z > z1-α, where:
z
ab
a s b s
2
2
b
2
2
a
Problem with the Sobel Test
• The distribution of ab is highly positively
skewed, and thus relying on a method that
assumes a normally distributed sampling
distribution (i.e., Sobel test) reduces power
– In other words, the tail probability is going to
be larger (larger p-value) if we assume a
normal distribution than if we assume a skewed
distribution
Bootstrapping
• Bootstrapping allows researchers the
opportunity to resample from the data in
order to generate an empirical sampling
distribution of ab
• Researchers can then use the empirical
(bootstrap) estimate of ab, along with the
standard error of the bootstrap estimates, to
compute a z statistic or a confidence interval
(which will generally be asymmetric)
Example
• Let’s say we are interested in determining if
anxiety (anx) mediates the relationship between
hindrances to doing well in a stats course (hindr)
and the exam average (exavg)
– r(anx,hindr)=.452*
– r(anx,exavg)=-.367*
– r(hindr,exavg)=-.303*
• Let’s use α = .05
Mediational Model
Anxiety
Hindrances
Exam
Average
Regression Equations
• Predicting exavg from hindr:
– Exavg = -3.874 (hindr) + 91.825 (p < .001)
• Predicting anx from hindr:
– Anx = .552 (hindr) + .335 (p < .001)
• Predicting exavg from both hindr and anx:
– Exavg = -2.203 (hindr) -3.027 (anx) + 94.352
– p-value for hindr = .062; p-value for anx = .002
• Therefore, since p(hindr)> α, many would say
(but we won’t) that anx fully mediates the
prediction of exavg from hindr
Added Variable Plots
• The relationship between examavg and hindrances
is weaker after controlling for anxiety
b=-3.87
b=-2.20
Sobel Test
• We can evaluate whether or not anx is a
significant mediator by using the Sobel test
– More specifically, we can evaluate the significance of
the indirect effect using the Sobel test
– We will need the regression coefficients (and standard
errors) for: 1) predicting anx from hindr and; 2)
predicting exavg from anx, with hindr in the equation:
• Anx = .335 + 0.552 (hindr)
– [SE (hindr) = .097]
• Exavg = 94.352 - 2.203 (hindr) - 3.027 (anx)
– [SE (anx) = .959]
Example
Sobel Test Calculation
z
ab
.552*30
. 27


27
.
6
22 22
2
2
2
2
asb bsa .552 *.959 (30
. 27) *.097
• Therefore, with Sobel z = -2.76 (p = .0058), we
can say that anxiety significantly mediates the
relationship between stats related hindrances and
the exam average
• A Sobel calculator is available at:
•http://www.afhayes.com/spss-sas-and-mplus-macros-and-code.html
Bootstrap Analyses
• Method 1: AMOS
• or any other SEM software that computes
bootstrapped indirect effects and standard errors
– Indirect Effect: -1.671
– Standard Error: .668
– z = -1.671/.668 = -2.50 (p = .0062)
Bootstrap Analyses
• Method 2: SPSS/SAS/MPlus Macros
• http://www.afhayes.com/spss-sas-and-mplusmacros-and-code.html
–
–
–
–
Indirect Effect: -1.683
95% CI: -3.145, -.5982
Standard Error: .678
z = -1.683/.678 = -2.48 (p = .0066)
Effect Size Measures
• Completely Standardized Indirect Effect
–
–
–
–
Product of the Beta coefficients for a and b
βa = .452
βb = -.289
CSIE = .452 * -.289 = -.131
• According to Cohen, .01-.09 is small, .10-.25 is
medium, and .25 + is large
– Thus we have a medium sized indirect effect
Effect Size Measures
• Proportion of the Total Effect that is
Mediated
– An alternative to the CSIE, that is very intuitive
• % Mediated = indirect effect / total effect = ab / c
– For our example:
• % Mediated = -1.671 / -3.874 = 43%
• Therefore, 43% of the relationship between
hindrances to doing well in stats and the exam
average is mediated by stats-related anxiety
Full Mediation
• There is a lot of debate over how to
determine full mediation, or whether full
mediation could ever exist in psychology
– Initially, it was suggested by Baron & Kenny that
a nonsignificant predictor effect, after controlling
for the mediator (path c΄), would indicate full
mediation
• However, statistical significance on its own is not
good, and with a small sample size, or a weak X-Y
relationship, we would find full mediation too often
Full Mediation, cont’d
• It has been suggested that a significant
Sobel test, and a % Mediated > .80, is
indicative of full mediation (davidakenny.net/cm/mediate.htm)
• Constance Mara derived an equivalenceand bootstrap-based test of full mediation
that involves demonstrating that the raw
correlation (rxy) is equivalent to the
correlation implied by the indirect effect
(rxy*) = βaβb
Multiple Mediators
• Many models evaluate the effects of more
than one mediator
• It is best to investigate all potential
mediators in one model so that you control
for the effects of the different mediators
• It is also advantageous to use statistical
software to conduct the separate
mediational analyses simultaneously
Mediational Model
Anxiety
Hindrances
Exam
Average
Exam
Ability
Multiple Mediator Model
in AMOS
• A disadvantage of AMOS is that generally
it only provides a total indirect effect, not
the separate indirect effects for each
mediator
• We can use “ghost variables” to be able to
obtain the separate indirect effect estimates
Multiple Mediational Model
in AMOS
Anxiety
b1
a1
Hindrances
a1
a2
Exam
Average
a2
b2
Exam
Ability
b1
b2
Bootstrap Analyses
• Method 1: AMOS
– Anxiety as Mediator:
– Indirect Effect: -1.813
– Standard Error: .670
– z = -1.859/.689 = -2.69 (p = .0036)
– Exam Ability as Mediator
– Indirect Effect: .265
– Standard Error: .244
– z = .265/.244 = 1.086 (p = .1387)
Bootstrap Analyses
• Method 2: SPSS Macro from Preacher/Hayes
– Anxiety as Mediator:
–
–
–
–
Indirect Effect: -1.859
95% CI: -3.3939, -.6915
Standard Error: .689
z = -1.859/.689 = -2.69 (p = .0036)
– Exam Ability as Mediator
–
–
–
–
Indirect Effect: .275
95% CI: -.0550, .9573
Standard Error: .247
z = .275/.247 = 1.113 (p = .1330)
Further Topics in Mediation
•
•
•
•
•
Latent Variable Mediation Models
Categorical Outcomes
Dichotomous or Multicategorical Predictors
Nonlinear relationships
4 variable mediation chains
– X  M1  M2  Y
Introduction to Moderation
• What is moderation? …. INTERACTION!
– The effect of X on Y differs at different levels
of the moderator
• Ex: The effects of provoked anger on aggression
differs at different levels of trait aggressiveness
– We could deal with interactions in ANOVA (all
categorical predictors), but ‘moderation’ often
implies interaction with at least one continuous
variable (and we would never categorize a
continuous variable in order to use ANOVA!)
Visual Representations
of Moderation
Mod
X
Y
X
Mod
X*Mod
Y
Understanding Moderation
• Interaction terms represent a relationship
between X and Y, and Mod and Y, that is
more than just additive
– In other words, the combined effect of X and
Mod is more substantial than what would be
expected by just summing the effects of X and
Mod
Two-way Model with Additive
Predictor Effects (i.e., no interaction)
High Mod
Medium Mod
Low Mod
Y
b0
X
Y΄ = β0 + β1 X + β2 Mod
Two-way Model with Interaction
High Mod
Med Mod
Low Mod
b0
X
Y΄ = β0 + β1X + β2Mod + β3X*Mod
Creating the Interaction Term
• The interaction term is created as the
product of the X and Mod variables
• Typically, continuous variables are mean
centered (i.e., X – M) and categorical
variables are dummy coded (e.g., 0,1)
before creating the interaction term
– Note that the main effect terms must be
included in the model with the interaction term,
or the interaction term is confounded
Centering
• Centering does not affect the interaction
term (β3)
• In fact, centering is not necessary for simply
screening for the presence of an interaction
• However, if you are going to interpret the
main effects with the interaction term in the
model (e.g., for conducting simple effects),
then the interaction term should be created
using appropriate centering
More on Centering
• When an interaction term is present in a model,
the effect of X is interpreted as the effect of X
when Mod = 0 (or the effect of Mod when X =
0), which is often uninformative
• Thus, we want to create an interaction term
which allows us to interpret the main effects in
a meaningful way, for example the effect of X
at the mean of Mod (which will be important
for understanding the nature of interactions)
Example: Two Continuous
Predictors
• We are interested in determining if
Classroom Independence (ci), Student
Independence (si), or the interaction
between them are important predictors of
grades in a class
– cic, sic are the mean-centred versions of si and
ci, and cic:sic (cic*sic) is the interaction term
• We start by exploring the interaction
Statistical Software Output
• Coefficients:
Estimate
Intercept 54.00
cic
.1733
sic
.1543
cic:sic .002852
Std. Error t value
4.991e-01 108.199
1.759e-02 9.851
1.711e-02 9.020
5.348e-04 5.334
Pr(>|t|)
< 2e-16
3.10e-16
1.90e-14
6.39e-07
- Since the interaction is significant, we want to
explore the nature of the interaction (and avoid
trying to interpret the main effects)
R Plot of the Interaction
Interpreting the Nature of the
Interaction
• As si increases, the relationship between ci and
grades gets stronger
• Although this R plot is very informative, often
researchers follow traditions such as creating a plot of
the effect of ci on grades at the mean and +/- one
standard deviation from the mean of si, and obtaining
the significance of the simple slopes
• These plots/analyses are done using the calculators at:
http://quantpsy.org/interact/mlr2.htm, and excel files are
available at: http://www.jeremydawson.co.uk/slopes.htm
Plot of the Interaction
• 1 sd below mean, mean, 1 sd above mean
Simple Slopes Analysis
• Simple Slopes at Conditional Values of SIC
– At 1 sd below the mean on SIC
• cic simple slope = 0.0991, t=3.379, p=0.0011
– At the mean on SIC
• cic simple slope = 0.1733, t=9.851, p<.0001
– At 1 sd above the mean on SIC
• cic simple slope = 0.2475, t=9.7616, p<.0001
Simple Slopes Analyses
• It is very easy to conduct simple slopes
analyses without available macros
• Since our interaction terms were created from centered
variables, the effect of cic on grades is at the mean of sic
(i.e., sic = 0) so that is our first simple slope
• We can obtain the simple slope of cic on grades at one sd
below the mean on sic by shifting our data so that sic = 0
is at one sd below the mean
– We do that by ADDING one sd to the scores on the centred si
variable before creating the interaction term
• We reverse this procedure for obtaining the simple slope
of ci on grades at one sd above the mean of si
More on Simple Slopes Analyses
• Simple slopes analyses are often over-used
– A significant interaction indicates that the effect
of X on Y differs across the levels of mod
– So, if you focus on the significance of the
simple slopes, you might be missing the
important point which is that the slopes differ
significantly (regardless of the statistical
significance of any particular simple slope)
• This is the same as focusing on simple effects when
following up an interaction in ANOVA
What if our Moderator is
Dichotomous?
• If our moderator is dichotomous, nothing
changes except that we dummy code (0,1)
the variable instead of mean centering it
– If the interaction is significant, we are now
interested in exploring the simple slopes of X
on Y at mod = 0 and mod = 1
• We do that by reverse coding the variable in order to
obtain both simple slopes
– As before we can graph and explain the nature
of the interaction
What if our moderator is
multicategorical?
• If our moderator has more than two
categories (c), then we need c-1 dummy
variables
– 3 Category Example (first category as referent):
• Dummy Variable 1 (DV1): c1=0, c2=1, c3=0
• Dummy Variable 2 (DV2): c1=0, c2=0, c3=1
– We now create 2 interaction variables:
• Int 1 = DV1 * X (note that X is mean centered)
• Int 2 = DV2 * X
What if our moderator is
multicategorical?
Y΄ = β0 + β1X + β2DV1+ β3DV2 + β4X*DV1+ β5X*DV2
• We include X, both dummy variables, and the
two interaction variables in our regression
equation
– The significance of the interaction is determined using
the R2 change from a hierarchical regression, where
the interaction terms are added to the model after the
main effects
What if our moderator is
multicategorical?
– To follow-up a significant interaction, it is again
important to plot the interaction
• The simple slopes from the plots are useful for
interpretation (i.e., understanding how the relationship
between X and Y differs for different levels of mod), but
the significance of the simple slopes is usually
uninformative
– Useful tests for interpreting the interaction are the
DV*X interactions, as each tests whether the slope
differs for the referent and the comparison groups
(and the referent group can be changed)
Higher-Order Interactions
• If you are interested in testing 3-way
interactions in regression (and I don’t
recommend testing 4-way or higher
interactions), all of the same centering,
coding, etc. rules apply
• The question being asked is whether the 2way interaction between X and Z differs
across the values of mod
Higher-Order Interactions
Y΄ = β0 + β1X + β2Z+ β3Mod + β4X*Z+ β5X*Mod + β6Mod*Z+ β7X*Z*Mod
– Interpreting 3-way interactions REQUIRES plots
– More specifically, start by plotting the X*Z
interaction at different levels of mod, and then
following up each as if it were a two-way
interaction
• There are different ways to plot the interaction, and the
method you choose will depend on how the variables are
coded and the theory being evaluated
• As always, try to focus less on the significance of the
simple slopes, or simple, simple slopes, and more on
interpreting the nature of the interaction
Further Topics in Moderation and
Mediation
•
•
•
•
•
Categorical outcome variables
Moderated Mediation
Mediated Moderation
Nonlinear interactions
Comparing Software for Analyzing
Mediation and Moderation
• Etc., Etc., Etc. ……